Fitting the Smile Revisited

Fitting the Smile Revisited: A Least Squares Kernel Estimator for the

Implied Volatility Surface

Matthias R. Fengler

Institut f¨ur ¨Okonometrie und Statistik, Wirtschaftswissenschaftliche Fakult¨at CASE – Center for Applied Statistics and Economics

Humboldt-Universit¨at zu Berlin, Spandauer Straße 1, 10178 Berlin, Germany

fengler@wiwi.hu-berlin.de TEL ++49 30 2093 5654 FAX ++49 30 2093 5649

Qihua Wang

Academy of Mathematics and System Science Chinese Academy of Science

Beijing, 100080 P.R. China

qhwang@mail.amss.ac.cn

August 6, 2003

∗I gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft and the Sonder- forschungsbereich 373 “Quantifikation und Simulation ¨okonomischer Prozesse”.

†The second author was supported by the national natural scientific foundation (key grant: 10231030).

Abstract

Nonparametric methods for estimating the implied volatility surface or the implied volatility smile are very popular, since they do not impose a specific functional form on the estimate. Traditionally, these methods are two-step estimators. The first step requires to extract implied volatility data from observed option prices, in the second step the actual fitting algorithm is applied. These two-step estimators may be se- riously biased when option prices are observed with measurement errors. Moreover, after the nonlinear transformation of the option prices the error distribution will be complicated and less tractable. In this study, we propose a one-step estimator for the implied volatility surface based on a least squares kernel smoother of the Black-Scholes formula. Consistency and the asymptotic distribution of the estimate are provided.

We demonstrate the estimator using German DAX index option data to recover the smile and the implied volatility surface.

Keywords: implied volatility surface, smile, Black-Scholes formula, least squares kernel smoothing

1 Introduction

Functional flexibility is the key challenge for model building and model selection in quanti- tative finance: often it is difficult, or sometimes impossible to justify on theoretical grounds a specific parametric form of the economic relationship under investigation. Furthermore, in a dynamic context, the economic structure may be liable to sizable changes and considerable fluctuation. Thus, estimation techniques that do not impose any a priori restrictions on the estimate, such as non- and semiparametric methods, are increasingly popular in financial practice.

Recently in finance, the implied volatility surface (IVS) has attracted much attention. Since it is derived from option prices observed at a certain point in time via the Black-Scholes (BS) formula, it is a widely accepted state variable that reflects current market expectations and market sentiments in a forward looking and instantaneous manner. Traders continuously monitor and update the IVS they ‘trade on’. Pricing and risk management tools are fed with the IVS currently prevailing in the market. In the case of the IVS, however, model flexibility is a necessity rather than an option: from the BS theory, the IVS should be a flat and constant function across strike prices K and the term structure τ of the option’s time to maturity. Yet, as a matter of fact, one observes rich functional patterns fluctuating through time, see Figure (1) for illustration. This dependence, captured by the function σb_t : (K, τ)→bσ_t(K, τ), is called IVS.

Parametric attempts to model the IVS along the strike profile, i.e. the ‘smile’, usually em- ploy quadratic specifications, Shimko (1993), An´e and Geman (1999), and Tompkins (1999) among others. However, it seems that these approaches are not capable of capturing the salient features of IVS patterns, and hence estimates may be biased. In a comprehensive study, Dumas et al. (1998) document this for a variety of parametric choices of local volatility models. In an attempt to allow for more flexibility, Hafner and Wallmeier (2001) fit quadratic splines to the smile function. A¨ıt-Sahalia and Lo (1998); Rosenberg (2000); A¨ıt-Sahalia et al.

(2001b); Cont and da Fonseca (2002); Fengler et al. (2003a) employ a Nadaraya-Watson esti- mator of the IVS function, and higher order local polynomial smoothing of the IVS is used in Rookley (1997). A¨ıt-Sahalia et al. (2001a) discuss model selection between fully parametric, semi- and nonparametric IVS specifications and argue in favor of the latter approaches. In

IVS on Jan. 02, 2001

0.13 0.21

0.29 0.37 0.46

0.71 0.88 1.05 1.22 1.39

0.21 0.25 0.30 0.34 0.38

Figure 1: IVS as observed on Jan. 02, 2001, calculated separately for each observed option price from the DAX Index options and discounted DAX Future prices. Lower front axis is moneyness κ_t = K/(F_te^−rτ) and lower right axis time to maturity measured in years corresponding to 17, 45, 73, and 164 days to expiry.

combining techniques from functional data analysis and backfitting of generalized additive models, Fengler et al. (2003b) propose a modeling approach of the IVS that fits only in the local neighborhood of the observed design points of the IVS. An estimate of the IVS of the particular day is then given by a sum of smooth basis functions, whose weights may change over time. The decisive advantage of this estimator is in the fact that it is tailored to the degenerated, i.e. discrete, ‘string’ data structure, which can be seen in Figure (1).

All these approaches share in common that they aretwo-step estimators by nature: in a first step, implied volatilities are derived by equating the BS formula with observed market prices and solving for the diffusion coefficient, Manaster and Koehler (1982); in the second step the actual fitting algorithm is applied. These two-step estimators may be seriously biased, when option prices or other input parameters can only be observed with errors. Moreover, the nonlinear transformation of option prices makes the error distribution less tractable.

Indeed, it has been conjectured that the presence of measurement errors can be of substantial impact, see Roll (1984), Harvey and Whaley (1991), and particularly Hentschel (2002) for

an extensive study on errors in implied volatility estimation and their possible magnitude.

Potential error sources are the bid-ask bounce, nonsynchronous pricing, infrequent trading of index stocks, and finite quote precision.

Here, we propose a one-step procedure based on a least squares kernel estimator that directly takes option prices, say calls Ce_t, and other variables observed at time t as input parameters:

σ(κb _t, τ) = arg min

σ n

X

i=1

n

Ce_ti−S_tiC^BS(·, σ)o2

W(κ_ti)K₁

κ_t−κ_ti h_1,n

K₂

τ −τ_i h_2,n

,

where C^BS(·, σ) denotes the BS price for calls, κt = K/St is moneyness, i.e. the rescaled strike dimension ‘strike by asset price’. K₁(·) and K₂(·) are kernel functions. W(·) denotes a uniformly continuous and bounded weight function, which allows for differential weights of observed option prices. These weights may be useful in the following respect: it is usually argued that in-the-money options contain a liquidity premium and should be incorporated to a lesser extent into the implied volatility estimate, or even excluded, A¨ıt-Sahalia and Lo (1998); Skiadopoulos et al. (1999). This goal may be achieved in using the weight function W(κ).

Our estimator can be interpreted as a localizedversion of estimation approaches in the early implied volatility literature, Latan´e and Rendelman (1976); Chiras and Manaster (1978);

Schmalensee and Trippi (1978); Beckers (1981); Whaley (1982). These studies are little concerned in obtaining a fit of the IVS as such, but rather in recovering one single, good predictor of future stock price variability: e.g. Whaley (1982) minimizes an equally weighted sum of squared pricing errors. This line of literature, however, fails to explore the asymptotic properties of their estimators.

We provide asymptotic results, and computation of confidence intervals around the estimates is straight forward. Inspecting confidence intervals of the fitted smile is notoriously neglected in the implied volatility literature, even when traditional two-step estimators are used. In the computation of our confidence bands the nonlinear inversion of option prices to derive implied volatility is explicitly taken into account. Thus they are more accurate than those obtained from two-step estimators. Confidence bands are critically important to traders, since they help them to discern for a given IVS estimate, whether new observations constitute a significant change of the IVS. Furthermore, trading models may be constructed in a way

such that the generation of trading signals is based on the confidence of the current IVS estimate.

The paper is structured as follows: in Section (2) the estimator in given in detail, consistency and the asymptotic distribution theory is established. Section (3) presents IVS estimates on German DAX index data, Section (4) concludes. Proofs are given in an appendix.

2 Least Squares Kernel Smoothing of the IVS

European style calls and puts are contingent claims on an assetS_t (for simplicity, paying no dividends, here), which yield as pay-off max(ST −K,0) and max(K−ST,0), respectively, at a given expiry dayT. K is called strike price. In the traditional BS model it is assumed that the asset price process S_t follows a geometric Brownian motion with a constant diffusion coefficient σ. Under no arbitrage, the BS option pricing formula for calls is given by (Black and Scholes, 1973):

C_t^BS(S_t, K, τ, r, σ) = S_tΦ(d₁)−Ke^−rτΦ(d₂), (1) d1 = ln(S_t/K) + (r+¹₂σ²)τ

σ√

τ ,

d₂ = d₁−σ√ τ ,

where Φ(·) denotes the cumulative distribution function of a standard normal random vari- able, rdenotes the risk-free interest rate, S_t the price of underlying at timet,τ =T−t time to maturity and K the exercise price. Put prices are derived from the put-call parity

C_t−P_t =S_t−e^−rτK . (2)

Empirically, the actual volatility σ of the underlying price process is the only parameter that cannot be observed directly. Hence one studies the volatility that is implied in option prices observed on markets, given the BS model were a true description of market conditions:

implied volatility σb is defined as that σ which equates observed market prices Ce_t with the theoretical BS option price, i.e.

bσ: C_t^BS(S_t, K, τ, r,bσ)−Ce_t= 0 . (3)

BS formula is monotone in the volatility parameterσ, which ensures uniqueness of a solution σ >b 0. The purpose will be to estimate IVS given byσbt : (K, τ)→bσt(K, τ).

On the trading floors, only options near at-the-money, ATM, i.e. K ≈S_t are traded at high liquidity. Since this region changes as the asset price fluctuates, it is convention to work on a moneyness metric, i.e. one scales strike prices by the current asset price (or future prices).

Denote moneyness by κt = K/St. Then ATM is around κt ≈ 1. A call option is called out-of-the-money, OTM, (in-of-the-money, ITM) if κ >1 (κ <1) with the reverse applying to puts.

Rewriting (1) in terms of moneyness yields

C_t^BS(S_t, K, τ, r, σ) =S_tC(κ_t, τ, r, σ) where C(κ_t, τ, r, σ) = Φ(d₁)− _S^K

te^−rτΦ(d₂) and d₁ = ^−lnκt+(r+

1 2σ²)τ σ√

τ , d₂ = d₁ −σ√

τ (as before).

Our least squares kernel estimator is defined by

bσ(κ_t, τ) = arg min

σ n

X

i=1

n

Ce_ti−S_tiC(κ_ti, τ_i, r_i, σ)o2

W(κ_ti)K₁

κ_t−κ_ti h1,n

K₂

τ −τ_i h2,n

, (4) where K₁(·) and K₂(·) are kernel functions and W(·) is a weight function. i= 1, . . . , n is a numbering of observed option prices.

We make the following assumptions:

(A1) ECe_t⁴ <∞, ES_t⁴ <∞ and EK⁴ <∞, whereE denotes the expectation operator.

(A2) W(·) is a uniformly continuous and bounded weight function.

(A3) K₁(·) and K₂(·) are bounded probability density kernel functions with bounded sup- port.

(A4) Interest rate r is a fixed constant.

Assumption (A1), can be considered as a general characterization of the framework in which computing the implied volatility is ‘sensible’. Clearly, it holds in the BS model, but also

for the classes of diffusion and affine jump-diffusion models studied in finance, see e.g Duffie et al. (2000). This is important, because practitioners like to use implied volatility in more general circumstances than the BS model provides. One may conjecture that EK⁴ <∞ is not needed becauseK is fixed in each option. However, when options are sampled randomly, it is useful to allow for this generalization. Note also thatECe_t⁴ <∞is actually an implication of ES_t⁴ < ∞ and EK⁴ < ∞, as long as we consider plain vanilla European options only.

Since put options have a bounded payoff, there is no need for an additional assumption.

Assumption (A2) is very common, and some important weight function satisfy it. In Sec- tion (3) we will discuss possible choices of W(·). (A3) is a condition met by a lot of kernels used in nonparametric regression, such as the quartic or Epanechnikov kernel functions. (A4) is an assumption often used in the option pricing literature including the BS model. It is gen- erally justified by the empirical observation that asset pricing variability largely outweighs changes in the interest rate. Nevertheless, the impact from changing interest rates can be substantial for options with a very long time to maturity. We also remark that including a fixed dividend yield into the model would be a straight forward extension of our estimator.

Given assumptions (A1) to (A4) we obtain consistency:

Theorem 1. Let σ(κ_t, τ) be the solution of E[{Ce_t1 −S_t1C(κ_t, τ, r₁, σ)}W(κ_t)] = 0. If con- ditions (A1), (A2), (A3) and (A4) are satisfied, we have

bσ(κ_t, τ)−→^p σ(κ_t, τ) as nh_1,nh_2,n → ∞.

The proof is contained in the appendix. For our next result, we introduce the notations A_i(κ_ti, τ_i, r_i, σ) ^def= Ce_ti −S_tiC(κ_ti, τ_i, r_i, σ),

B(κ_t, τ, r, σ) ^def= ∂C(κ_t, τ, r, σ)

∂σ =S_t⁻¹∂C^BS(·)

∂σ =√

τ φ(d₁), (5)

D(κ_t, τ, r, σ) ^def= ∂²C(κ_t, τ, r, σ)

∂²σ =S_t⁻¹∂²C^BS(·)

∂σ² =√

τ φ(d₁)d₁d₂σ⁻¹ , (6) where φ(u) = ^√1_2πe^−u2/2. In financial language, B, the sensitivity of the option price with respect to implied volatility changes is called the option ‘vega’. It is an important quantity for portfolio managers wishing to hedge their volatility risk. The second derivative Dis also

referred to as option ‘volga’. From the put-call parity in (2) it is seen that vega and volga are identical for calls and puts.

We establish

Theorem 2. Under conditions (A1), (A2), (A3), and (A4) if

E{B²(κ_t, τ, r, σ)S_t²W(κ_t)} 6=E{A(κ_t, τ, r, σ)D(κ_t, τ, r, σ)S_tW(κ_t)}, we have pnh1nh2n{σ(κb t, τ)−σ(κt, τ)}−→ N^L (0, γ⁻²ν²), where

γ^{2 def}= h

E{−B²(κt, τ, r, σ)S_t²W(κt) +A(κt, τ, r, σ)D(κt, τ, r, σ)StW(κt)}i2

f(κt, τ) ν^{2 def}= E{A²(κ_t, τ, r, σ)B²(κ_t, τ, r, σ)S_t²W²(κ_t)}

Z

K₁²(u)K₂²(v)dudv , and f(κ_t, τ) is the joint probability density function of κ_t and τ respectively.

The proof is again given in the appendix. Finally, the results carry over to put options:

Corollar 1. By the put-call-parity and the bounded pay-off of put options, Theorem (1) and (2), hold also for put options, with A replaced correspondingly.

The asymptotic distribution intricately depends on first and second order derivatives, and the particular weight function. Nevertheless an approximation is simple, since first and second order derivatives have the analytical expressions given in (5) and (6).

3 Applications

3.1 Weighting Functions, Kernels, and Minimization Scheme

In the vain of obtaining a good forecast of asset price variability, the early literature on implied volatility discusses weighting the observations intensively. Schmalensee and Trippi (1978) and Whaley (1982) argue in favor of unweighted averages, i.e. they use scalar estimate

bσ^∗ = arg min

σ n

X

i=1

{(Ce_i−C(σ)}² , (7)

as a predictor of the future stock variability. Beckers (1981) minimizes

bσ^∗ = arg min

σ N

X

i=1

w_i{Ce_i−C(σ)}²/

N

X

i

w_i , (8)

where w_i ^def= ∂C_i/∂σ, the option vega. Similarly, Latan´e and Rendelman (1976) use the squared vega as weights:

bσ^∗ = v u u t

N

X

i=1

w_i²bσ²/

N

X

i

w_i , (9)

Finally, Chiras and Manaster (1978) propose to employ the elasticity with respect to volatil- ity:

bσ^∗ =

N

X

i=1

η_ibσ/

N

X

i

η_i , (10)

where ηi

def= ^∂C_∂σⁱ_C^σ

i.

For calls and puts, vega is Gaussian shaped function in the underlying centered (roughly) ATM, Equation (5). Elasticity is a decreasing (increasing) function in the underlying for calls (puts). Common concern of the weighting procedures is to give low weight to ITM options, and highest weight to ATM or OTM options: ITM options are more expensive than ATM and OTM options because their intrinsic value, i.e. their payoff function evaluated at the current underlying prices, is already positive. Thus, they provide lower leverage for speculation, and produce higher costs in portfolio hedging. Due to their lower trading volume, they are suspected to sell at a liquidity premium which may ensue biased estimates of implied volatility. Consequently, some authors delete or downweigh ITM options, A¨ıt-Sahalia and Lo (1998); Skiadopoulos et al. (1999).

Our estimator is general enough to allow for uniformly continuous and bounded weighting functionsW(κ) depending on moneyness. Technically, it is possible to use weights depending also on other variables includingσas done in (8) to (10). For several reasons, however, we like to refrain from using more involved weight functions: first, when ITM options are deleted or downweighted in the more recent literature, this choice is entirely determined by moneyness, not by vega. From this point of view, to have the weighting scheme depend on σ is rather implicit. Second, from a statistical point of view, weights depending on σ are likely to blow up the asymptotic variances in from of derivatives of W. This complicates estimation

and computation of confidence bands without adding to the problem of recovering a good estimate of the implied volatility surface. Finally, if one likes weights looking like the option vega or elasticity with respect to volatility, one may very easily construct weights W(κ) that look very similar. For instance, an estimator in the type of Latan´e and Rendelman (1976) would put W shaped as a Gaussian density.

For the purpose of IVS estimation in our particular application, we like to give less weight to ITM options. This can be achieved by using as weighting functions

W(κ) = 1

π arctann

α(1−κ)o

+ 0.5, for calls, and for puts

W(κ) = 1

π arctan n

α(κ−1) o

+ 0.5,

where π = 3.141... is the circle constant. α controls the speed, with which ITM options receive lower weight. ATM options are equally weighted. Outside κ≈1, only OTM options enter the minimization at significant weight. In our application we choose α = 9. Other values are perfectly possible, and this choice is motivated to have a gentle transition between OTM call and OTM put options. The ultimate choice of α will depend on the specific application at hand.

The kernel functions employed throughout are quartic kernels, i.e.

K(u) = 15

16(1−u²)²I(|u| ≤1),

where I(A) denotes the indicator function of the event A. Other bounded kernels may be used, such as the Epanechnikov kernel. In practice, choice of the kernel functions has little impact on the estimates, H¨ardle (1990). Since the minimization is globally concave (compare proof of Theorem (1)), and well posed as long as h₁ and h₂ do not become unreasonably small, any minimization algorithm for globally concave objective functions can be employed.

We use the Golden section search, Press et al. (1993), which is implemented in XploRe, www.i-xplore.de. The tolerance, i.e. the fractional precision of the minimum, is maintained at 10⁻⁸.

Observation Time to min max mean standard total number of

date expiry (days) deviation observations calls

Jan. 02, 2001 17 0.1711 0.3796 0.2450 0.0190 1219 561

45 0.2112 0.2839 0.2425 0.0169 267 134

73 0.1951 0.3190 0.2497 0.0199 391 209

164 0.1777 0.3169 0.2528 0.0229 178 76

Feb. 02, 2001 14 0.1199 0.4615 0.1730 0.0211 1560 813

42 0.1604 0.2858 0.1855 0.0188 715 329

77 0.1628 0.2208 0.1910 0.0172 128 45

133 0.1645 0.2457 0.1954 0.0221 119 63

Table 1: Implied volatility data as obtained by inverting the BS formula separately for each observation in the sense of two-step estimators.

3.2 Data Description and Estimations

The data used contains tick statistics on the DAX future and DAX index options and is provided by the German-Swiss Futures Exchange EUREX for January and February 2001.

Both future and option data are contract based data, i.e. each single contract is registered together with its price, contract size, and time of settlement up to a hundredth second.

Interest rate data in daily frequency, i.e. 1, 3, 6, 12 months EURIBOR rates, is gathered from Thomson Financial Datastream, and linearly interpolated to approximate the riskless interest rate for the option specific time to maturity.

For our application, we use data from January 02 and February 02, 2001. In a first step, we recover the DAX index values. To this end, we group to each option price observation C_t or P_t the future price F_t of the nearest available future contract, which was traded within a one minute interval around the observed option. The future price observation was taken from the most heavily traded future contract on the particular day, which is the March 2001 contract. The no-arbitrage price of the underlying index in a frictionless market without dividends is given by S_t=F_te^{−rTF ,t(TF−t)}, where S_t and F_t denote the index and the future price respectively, T_F the future’s maturity date, and r_T,t the interest rate with maturity T −t.

0.9 1 1.1 1.2 Moneyness

0510

Figure 2: Nonparametrically estimated densities of observed moneyness κ_t =K/(F_te^−rτ)for Jan. 02, 2001, and options with 17 days to expiry. Solid line for all observations, thickly dashed line for puts, and the more thinly dashed line for calls only. Quartic kernel used, bandwidth chosen according to Silverman’s rule of thumb. Density plots for other expiry dates including Feb. 02., 2001, available upon request.

In the case of an capital weighted performance index as is the DAX index, Deutsche B¨orse (2002), dividends less corporate tax are reinvested into the index. Thus, dividends should not play any role for option valuation. In practice, however, one still observes – especially in late spring and early summer during the ‘dividend season’ of DAX index companies – deviances between put and call implied volatility, when the discounted futures are used as the underlying asset. For this case, a correction schemed is developed by Hafner and Wallmeier (2001), also described in Fengler et al. (2003b). In our case, this problem is mitigated by the fact that we use January and February data. Furthermore in using a weight function that downweighs ITM options, which are most sensitive to this dividend wedge, this correction can also be achieved by the weighting function W itself.

In Table (1) we give an overview of the data employed. We prefer to present summary statis- tics in the form of implied volatility data obtained by inverting the BS formula separately for each observation rather than the option price data itself. The corresponding option data

can be seen in the top panel of Figure (3). Since the data is transaction data containing potential misprints, we also applied a filter in deleting all observations whose implied volatil- ity is bigger than 0.7 and less than 0.1. For the distribution of the data across moneyness compare Figure (2) which presents density plots of moneyness for calls, puts, and all the observations observed on Jan.02 for 17 days to expiry. Put and call densities appear shifted which is due to the aforementioned liquidity argument of ATM and OTM options. For sake of space we do not present the very similar plots for the other expiry dates and Feb. 02, 2001. The settlement price of the March 2001 future was 6340 EUR at a volume of 30 088 contracts on Jan. 02, 2001, and 6669.5 EUR and 34 244 contracts on Feb. 02, 2001.

For our smile fits we pick the shortest time to expiry options of the Jan. 02 and Feb. 02, 2001, data. Plots are displayed in Figures 3 and 4. The top panel shows observed option prices given on the moneyness scale, while the lower panel demonstrates the estimate. In Figure 5, fits for the entire IVS are presented.

4 Conclusion

In this paper we present a new estimation approach to fit implied volatility smiles and surfaces. Usually one inverts first the Black-Scholes formula to back out implied volatility data and applies the fitting algorithm in a second step. This procedure may yield biased estimates when option price data is observed with measurement error only. Here, we provide a one-step estimator of the implied volatility surface. Our estimator is a least squares kernel estimator which directly smoothes implied volatility in the option price space. Asymptotic theory for the estimator is provided. We demonstrate the estimator using German DAX index option data to recover the smile and the implied volatility surface.

0.9 1 1.1 1.2 Moneyness

0200400600

0.9 1 1.1 1.2

Moneyness

0.150.20.250.30.350.40.45

Figure 3: Upper panel: Observed option price data on Jan 02, 2001. From lower left to upper right put prices, from upper left to lower right call prices. Lower panel: Least squares kernel smoothed implied volatility smile for 17 days to expiry on Jan 02, 2001. Bandwidth h₁ = 0.025, quartic Kernels employed. Minimization achieved by Golden section search.

Dotted lines are the 95% confidence intervals for bσ. Single dots are implied volatility data obtained by inverting the BS formula separately for each observation in the sense of two-step

0.85 0.9 0.95 1 1.05 1.1 1.15 Moneyness

05001000

0.85 0.9 0.95 1 1.05 1.1 1.15

Moneyness

0.10.20.30.4

Figure 4: Upper panel: Observed option price data on Feb. 02, 2001. From lower left to upper right put prices, from upper left to lower right call prices. Lower panel: Least squares kernel smoothed implied volatility smile for 14 days to expiry on Jan 02, 2001. Bandwidth h₁ = 0.015, quartic Kernels employed. Minimization achieved by Golden section search.

Dotted lines are the 95% confidence intervals for bσ. Single dots are implied volatility data obtained by inverting the BS formula separately for each observation in the sense of two-step

IVS on Jan. 02, 2001

0.08 0.12

0.16 0.20

0.24

0.78 0.88

0.98 1.08

1.18 0.21

0.25 0.30 0.34 0.38

IVS on Feb. 02, 2001

0.08 0.12

0.15 0.19

0.23

0.75 0.84

0.93 1.02

1.11 0.19

0.26 0.32 0.39 0.46

Figure 5: Left panel: IVS fit for Jan. 02, 2001, with least squares kernel smoother. Band- widths are h1 = 0.05 in moneyness direction and h2 = 0.10 in time to maturity direction.

Right panel: IVS fit for Feb. 02, 2001, with least squares kernel smoother. Bandwidths are h₁ = 0.03andh₂ = 0.07. In both panels, single dots denote implied volatility data obtained by inverting the BS formula separately for each observation in the sense of two-step estimators.

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A Proofs

Proof of Theorem (1):

Let

Lb_n(σ)^def= 1 nh_1,nh_2,n

n

X

i=1

{Ce_ti−S_tiC(κ_ti, τ_i, r_i, σ)}²W(κ_ti)K₁

κ_t−κ_ti h_1,n

K₂

τ −τ_i h_2,n

.

As a first step, let us prove

Lb_n(σ)−→^p L(σ)^def= Eh

{Ce_t−S_tC(κ_t, τ, r;σ)}²W(κ_t)i

. (11)

It is observed that Lb_n(σ) = 1

nh_1,nh_2,n

n

X

i=1

{Ce_ti −S_tiC(κ_ti, τ_i, r_i;σ)}²W(κ_ti)K₁

κ_t−κ_ti h_1,n

K₂

τ−τ_i h_2,n

−E

{Ceti −StiC(κti, τi, ri;σ)}²W(κti)K1

κ_t−κ_ti h_1,n

K2

τ−τ_i h_2,n

+ 1

h_1,nh_2,nE

{Ce_t1 −S_t1C(κ_t1, τ₁, r₁;σ)}²W(κ_t1)K₁

κ_t−κ_t1 h_1,n

K₂

τ−τ₁ h_2,n

def= α_n+β_n (12)

Standard arguments can be used to prove

Eα²_n=O((nh_1,nh_2,n)⁻¹) (13) by conditions (A1) and (A2).

By Taylor’s expansion, we have

β_n = 1

hn,1hn,2

E Z

{Ce_t1 −S_t1C(x, y, r₁;σ)}²W(x)K₁

κ_t−x h1,n

K₂

τ −y h2,n

dx dy

= E

Z

{Ce_t−S_tC(κ_t−h_nu, τ −h_nv, r;σ)}²W(κ_t−h_nu)K₁(u)K₂(v)du dv

−→p L(σ). (14)

(13) and (14) together prove (11).

In a second step, we have, recalling the definition of σ(κ_t, τ)

∂L(σ)

∂σ

σ=σ(κt,τ) = −2ECe_tS_tW(κ_t) ∂

∂σC(κ_t, τ, r, σ)

σ=σ(κt,τ)

+2ES_t²C(κ_t, τ, r, σ(κ_t, τ))W(κ_t) ∂

∂σC(κ_t, τ, r, σ)

σ=σ(κt,τ) = 0 and

∂²L(σ)

∂σ²

σ=σ(κt,τ)

= −2ECetStW(κt) ∂²

∂σ²C(κt, τ, r, σ)

σ=σ(κt,τ)

+2ES_t²W(κ_t) ∂

∂σC(κ_t, τ, r, σ) _σ=σ(κ

t,τ)

2

+2ES_t²W(κ_t)C(κ_t, τ, r, σ(κ, τ)) ∂²

∂σ²C(κ_t, τ, r, σ)

σ=σ(κ,τ)

= 2ES_t²W(κt) ∂

∂σC(κt, τ, r, σ)

σ=σ(κt,τ)

2

.

This together with (11) proves that Lbn(σ) converges in probability to a concave function with a unique minimum at σ =σ(κt, τ). Thus bσn(κt, τ)−→^p σ(κt, τ) is proved.

Proof of Theorem (2):

Recalling the definition of bσ(κ_t, τ), it follows that bσ(κ_t, τ) is the solution of the following equation

U_n(σ) ^def= (nh_1nh_2n)⁻¹

n

X

i=1

S_tiA_i(κ_ti, τ_i, r_i, σ)B_i(κ_ti, τ_i, r_i, σ)W(κ_ti)K₁

κ_t−κ_ti h_1,n

K₂

τ−τ_i h_2,n

= 0.

By Taylor’s expansion, we get

0 =U_n(σ(κb _t, τ)) =U_n(σ(κ_t, τ)) +U_n⁰(σ^∗)(bσ_t(κ_t, τ)−σ(κ_t, τ), (15) where σ^∗ lies between σ and σb and U_n⁰(σ^∗)^def= _∂σ^∂ U_n(σ)|_σ=σ^∗. By (15), we have

bσ(κ_t, τ)−σ(κ_t, τ) = −[U_n⁰(σ^∗)]⁻¹U_n(σ). (16)

By some algebra, we obtain U_n⁰(σ) = 1

nh_1,nh_2,n

n

X

i=1

nh ∂

∂σA_i(κ_ti, τ_i, r_i, σ)

B_i(κ_ti, τ_i, r_i, σ) +A_i(κ_ti, τ_i, r_i, σ) ∂

∂σB_i(κ_ti, τ_i, r_i, σ)i

S_tiW(κ_ti)

×K₁

κ_t−κ_ti hn

K₂

τ −τ_i h2n

−Eh

∂

∂σA_i(κ_ti, τ_i, r_i, σ)

B_i(κ_ti, τ_i, r_i, σ) (17) +A_i(κ_ti, τ_i, r_i, σ) ∂

∂σB_i(κ_ti, τ_i, r_i, σ)i

S_tiW(κ_ti)

×K₁

κ_t−κ_ti h1,n

K₂

τ −τ_i h2,n

io

+ 1

nh_1nh_2n

n

X

i=1

Enh

∂

∂σA_i(κ_ti, τ_i, r_i, σ)

B_i(κ_ti, τ_i, r_i, σ) +A_i(κ_ti, τ_i, r_i, σ) ∂

∂σB_i(κ_ti, τ_i, r_i, σ)i

S_tiW(κ_ti)

×K₁

κ_t−κ_ti hn

K₂

τ −τ_i h2n

o

def= 4_n1+4_n2 (18)

Inspect first 4_n1 in Equation (18): by some algebra, we get E4²_n1

≤ 1

n²h²_1,nh²_2,n

n

X

i=1

En ( ∂

∂σA_i(κ_ti, τ_i, r_i, σ))B_i(κ_ti, τ_i, r_i, σ) +A_i(κ_ti, τ_i, r_i, σ) ∂

∂σB_i(κ_ti, τ_i, r_i, σ)

S_tiW(κ_ti)

×K₁

κ_t−κ_ti h_n

K₂

τ −τ_i h_2n

o2

= f²(κ_t, τ)R

K₁²(u)duR

K₂²(v)dv

nh_1,nh_2,n Enh

∂

∂σA₁(κ_t, τ, r₁, σ)B₁(κ_t, τ, r₁, σ) +A₁(κ_t, τ, r₁, σ) ∂

∂σB₁(κ_t, τ, r, σ) 2

i

S_t²W(κ_t)o +O

1 nh_1,nh_2,n

−→0. (19) as nh_1,nh_2,n → ∞. f(κ_t, τ) denotes the joint probability density functions of κ_t and τ.

To consider 4_n2 in Equation (18), denote D(κ_t, τ, r, σ) = _∂σ^∂ B(κ_t, τ, r, σ), for simplicity.

Note that _∂σ^∂ A(κt, τ, r, σ) = −StB(κt, τ, r, σ). Thus, we have 4_n2 = 1

h_1nh_2nE Z

−B²(x, y, r, σ) +A(x, y, r, σ)D(x, y, r₁, σ)

S_tW(κ_t)

×K₁

κ_t−x h_1,n

K₂

κ_t−x h_2,n

f(x, y)dx dy]

= E

Z

[−B²(κ_t−h_1,nu, τ −h_2,nv, r₁, σ)S_t

+A(κ_t−h_1nu, τ −h_2,nv, r₁, σ)D(κ_t−h_1nu, τ −h_2,nv, r₁, σ)]S_tW(κ_t)

×f₁(κ_t−h_1,nu, τ −h_2,nv)K₁(u)K₂(v)du dv

−→ n

−E[B²(κ_t, τ, r, σ)S_t²W(κ_t)] +E[A(κ, τ, r₁, σ)D(κ, τ, r₁, σ)S_tW(κ_t)]o

f(κ_t, τ) (20) (18), (19), (20) and the fact U_n⁰(σ^∗)−U_n⁰(σ)→0 together prove

U_n⁰(σ^∗)−→^p n

E[−B²(κ_t, τ, r, σ)S_t²W(κ_t)] +E[A(k, τ, r, σ)D(κ, τ, r, σ)S_tW(κ_t)]o

f(κ_t, τ). (21)

Now, let

u_ni ^def= 1

h_1,nh_2,nA(κ_ti, τ_i, r_i, σ)B(κ_ti, τ_i, r_i, σ)S_tiW(κ_ti)K₁

κ−κ_ti h_1,n

K₂

τ −τ_i h_2.n

.

For some δ >0, we have E|u_ni|^2+δ = 1

h^2+δ_1,n h^2+δ_2,n EA^2+δ(κ_ti, τ_i, r_i, σ)B^2+δ(κ_ti, τ_i, r_i, σ)S_t^2+δ

i W^2+δ(κ_ti)

×K₁^2+δ

κ_t−κ_ti h_1,n

K₂^2+δ

τ−τ_i h_2,n

= 1

h^1+δ_1,n h^1+δ_2,n E Z

A^2+δ(κ_t−h_nu, τ −h_nv, r, σ)B^2+δ(κ_t−h_nu, τ −h_nu, r, σ)

×S_t^2+δW^2+δ(κ_t−h_nu)K₁^2+δ(u)K₂^2+δ(v)du dv

= f(κt, τ)R

K₁^2+δ(u)duR

K₂^2+δ(v)dv h^1+δ_1,nh^2+δ_2,n E

A^2+δ(κ_t, τ, σ)B^2+δ(κ_t, τ, σ)S_t^2+δW^2+δ(κ_t)

+^O 1

h^1+δ_1,n h^1+δ_2,n

!

. (22)

Similarly, we get

Eu²_ni = f(κ_t, τ)R

K₁²(u)duR

K₂²(v)dv

h_1,nh_2,n E{A²(κt, τ, r, σ)B²(κ, τ, r, σ)S_t²W²(κt)}

+^O

1 h_1,nh_2,n

. (23)

(22) and (23) together prove Pn

i=1E|u_ni|^2+δ (Pn

i=1E|u_ni|²)^2+δ2

=O((nh_1,nh_2,n)^−2δ) = O(1) as nh_1,nh_2,n →0.

Applying the Liapounov central limit theorem, we get

pnh_1,nh_2,nU_n(σ)−→ N^L (0, f(κ_t, τ)ν²), (24) where ν² is defined in Theorem (2).

By (21) and (24), Theorem (2) is proved.